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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdlsp | Structured version Visualization version GIF version |
Description: Span in the set of functionals with closed kernels. (Contributed by NM, 28-Mar-2015.) |
Ref | Expression |
---|---|
lcdlsp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcdlsp.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcdlsp.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcdlsp.m | ⊢ 𝑀 = (LSpan‘𝐷) |
lcdlsp.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcdlsp.f | ⊢ 𝐹 = (Base‘𝐶) |
lcdlsp.n | ⊢ 𝑁 = (LSpan‘𝐶) |
lcdlsp.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcdlsp.g | ⊢ (𝜑 → 𝐺 ⊆ 𝐹) |
Ref | Expression |
---|---|
lcdlsp | ⊢ (𝜑 → (𝑁‘𝐺) = (𝑀‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdlsp.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝐶) | |
2 | lcdlsp.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2731 | . . . . . 6 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
4 | lcdlsp.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
5 | lcdlsp.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | eqid 2731 | . . . . . 6 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
7 | eqid 2731 | . . . . . 6 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
8 | lcdlsp.d | . . . . . 6 ⊢ 𝐷 = (LDual‘𝑈) | |
9 | lcdlsp.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | lcdval 40923 | . . . . 5 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
11 | 10 | fveq2d 6895 | . . . 4 ⊢ (𝜑 → (LSpan‘𝐶) = (LSpan‘(𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))) |
12 | 1, 11 | eqtrid 2783 | . . 3 ⊢ (𝜑 → 𝑁 = (LSpan‘(𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))) |
13 | 12 | fveq1d 6893 | . 2 ⊢ (𝜑 → (𝑁‘𝐺) = ((LSpan‘(𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))‘𝐺)) |
14 | 2, 5, 9 | dvhlmod 40444 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
15 | 8, 14 | lduallmod 38486 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
16 | eqid 2731 | . . . 4 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷) | |
17 | eqid 2731 | . . . 4 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} | |
18 | 2, 5, 3, 6, 7, 8, 16, 17, 9 | lclkr 40867 | . . 3 ⊢ (𝜑 → {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ (LSubSp‘𝐷)) |
19 | lcdlsp.g | . . . 4 ⊢ (𝜑 → 𝐺 ⊆ 𝐹) | |
20 | lcdlsp.f | . . . . 5 ⊢ 𝐹 = (Base‘𝐶) | |
21 | 2, 3, 4, 20, 5, 6, 7, 17, 9 | lcdvbase 40927 | . . . 4 ⊢ (𝜑 → 𝐹 = {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) |
22 | 19, 21 | sseqtrd 4022 | . . 3 ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) |
23 | eqid 2731 | . . . 4 ⊢ (𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) = (𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) | |
24 | lcdlsp.m | . . . 4 ⊢ 𝑀 = (LSpan‘𝐷) | |
25 | eqid 2731 | . . . 4 ⊢ (LSpan‘(𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) = (LSpan‘(𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) | |
26 | 23, 24, 25, 16 | lsslsp 20858 | . . 3 ⊢ ((𝐷 ∈ LMod ∧ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ (LSubSp‘𝐷) ∧ 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) → (𝑀‘𝐺) = ((LSpan‘(𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))‘𝐺)) |
27 | 15, 18, 22, 26 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝑀‘𝐺) = ((LSpan‘(𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))‘𝐺)) |
28 | 13, 27 | eqtr4d 2774 | 1 ⊢ (𝜑 → (𝑁‘𝐺) = (𝑀‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {crab 3431 ⊆ wss 3948 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 ↾s cress 17180 LModclmod 20702 LSubSpclss 20774 LSpanclspn 20814 LFnlclfn 38390 LKerclk 38418 LDualcld 38456 HLchlt 38683 LHypclh 39318 DVecHcdvh 40412 ocHcoch 40681 LCDualclcd 40920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38286 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-mre 17537 df-mrc 17538 df-acs 17540 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cntz 19229 df-oppg 19258 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20946 df-lsatoms 38309 df-lshyp 38310 df-lcv 38352 df-lfl 38391 df-lkr 38419 df-ldual 38457 df-oposet 38509 df-ol 38511 df-oml 38512 df-covers 38599 df-ats 38600 df-atl 38631 df-cvlat 38655 df-hlat 38684 df-llines 38832 df-lplanes 38833 df-lvols 38834 df-lines 38835 df-psubsp 38837 df-pmap 38838 df-padd 39130 df-lhyp 39322 df-laut 39323 df-ldil 39438 df-ltrn 39439 df-trl 39493 df-tgrp 40077 df-tendo 40089 df-edring 40091 df-dveca 40337 df-disoa 40363 df-dvech 40413 df-dib 40473 df-dic 40507 df-dih 40563 df-doch 40682 df-djh 40729 df-lcdual 40921 |
This theorem is referenced by: lcdlkreqN 40956 mapdhvmap 41103 |
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